Equivalent condition for continuity in topological space.

Question

In metric spaces, we have this condition that a map $f$ is continuous at a point $x$ iff for each seq. $(X_n)$ converging to $x$ the seq. $(f(X_n))$ converges to $f(X)$. Is the same true in case of a topological space? i.e. is it true, A map $f$ on a topological space is continuous at a point $x$ iff for each net $(X_n)$ converging to $x$ the net $(f(X_n))$ converges to $f(x)$?

It's not true for sequences. See this post. – David Mitra Feb 01 '13 at 16:24 — David Mitra, Feb 01 '13 at 16:24
See: Elementary Topology, S 6.6, Proposition 10. – M.Sina Feb 01 '13 at 16:25 — M.Sina, Feb 01 '13 at 16:25

score 1 · Accepted Answer · answered Feb 01 '13 at 16:14

1

Yes, this is true. This is more or less the point of nets. It's also true if you replace nets with filters.

answered Feb 01 '13 at 16:14

Chris Eagle

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Equivalent condition for continuity in topological space.

1 Answers1